Estimation of latent waiting and service times from incomplete event logs

ABSTRACT

Methods and systems for estimating latent service and latent wait times include extracting transition times between activities from a partial event log. Parameters for respective gamma distributions are estimated that characterize latent waiting time and latent service time for each activity. A latent waiting time and latent service time for each activity is estimated based on the estimated parameters using a processor.

BACKGROUND Technical Field

The present invention relates to process analysis and, moreparticularly, to estimating waiting and service times from incompleteevent logs.

Description of the Related Art

Performance analysis is frequently used in redesigning processes toincrease their efficiency. One frequently used type of analysis iscalculation of waiting time and service time of process activities froman event log. The average waiting and service times of individualactivities and resources enable the redesigner to discover bottlenecksin the process.

Process management tools (e.g., a process aware information system(PAIS)) will record event details. For example, a PAIS may record bothstart-event and end-event activity for a variety of events. This makescalculating service time and waiting time trivial, simply by subtractingevent timestamps.

However, it is not always possible to access both start- and end-eventinformation. In many legacy event logs, only one of the two types ofinformation will be recorded. One example of such an event log is in,for example, webserver logs, which record only the start time of anevent and do not record the time at which the event completes.

In such a case, the only information available is the transition timefrom one event to the next (i.e., the time between respectivestart-event timestamps or respective end-event timestamps). While moremodern process managers will handle both, some customers may wish tohave analyses of process timing to improve their existing, legacysystems.

SUMMARY

A method for estimating latent service and latent wait times includesextracting transition times between activities from a partial event log.Parameters for respective gamma distributions are estimated thatcharacterize latent waiting time and latent service time for eachactivity. A latent waiting time and latent service time for eachactivity is estimated based on the estimated parameters using aprocessor.

A method for estimating latent service and latent wait times includesextracting transition times between activities from a partial event logthat comprises only one of start-event information and end-eventinformation for each activity. Parameters for respective gammadistributions are estimated that characterize latent waiting time andlatent service time for each activity. Estimating the parametersincludes determining a likelihood that the estimated parameters wouldreproduce the partial event log and iterating the estimation ofparameters and the determination of the likelihood until the likelihoodconverges. A latent waiting time and latent service time for eachactivity is estimated based on the estimated parameters using aprocessor.

A system for estimating latent service and latent wait times includes atransition time module configured to extract transition times betweenactivities from a partial event log. A parameter module is configured toestimate parameters for respective gamma distributions that characterizelatent waiting time and latent service time for each activity. A latenttime module having a processor is configured to estimate a latentwaiting time and latent service time for each activity based on theestimated parameters.

These and other features and advantages will become apparent from thefollowing detailed description of illustrative embodiments thereof,which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The disclosure will provide details in the following description ofpreferred embodiments with reference to the following figures wherein:

FIG. 1 is a diagram showing a timeline of activity execution inaccordance with the present principles;

FIG. 2 is a diagram showing a timeline of activity execution inaccordance with the present principles;

FIG. 3 is a block/flow diagram of a method for optimizing a process inaccordance with the present principles;

FIG. 4 is a block/flow diagram of a method for extracting transitiontimes from a partial event log in accordance with the presentprinciples;

FIG. 5 is a block/flow diagram of a method for determining latentservice and waiting times in accordance with the present principles; and

FIG. 6 is a block diagram of a latent time estimation system inaccordance with the present principles.

DETAILED DESCRIPTION

Embodiments of the present principle access transition times betweenevents from an incomplete log that lacks either event start times orevent end times. The present embodiments characterize the distributionof time durations as a gamma distribution and estimates a latent waitingtime and a latent service time for each transition.

Referring now to FIG. 1, an exemplary event timeline 100 is shown. Thistimeline 100 represents an ideal set of circumstances, where the top 102of the timeline shows events that are observable event and performanceindicators, the bottom 106 of the timeline shows unobservableperformance indicators, and the timeline itself 104 shows the recordedevents in the log. A series of activities (A, B, C) are recorded on thetimeline 104, with each having an “assign” event 108, marking the pointin time when a request for the activity was received, a “start” event110, marking the point in time when the event actually begins, and a“complete” event 112, marking the point in time when the eventcompletes.

Because so much information was collected, a time for the transitionsbetween activities and a time for the duration of an activity can beobserved on the top portion of the timeline 102. The duration of anactivity in this view includes a waiting time, representing the timethat expires between the request 108 for the activity and the time theactivity begins 110. The duration also includes a service time, whichrepresents the time from the start of the activity 110 to the completionof the activity. The duration is then followed by a transition period,until the next activity is assigned.

While the events recorded on the timeline 104 are available, they maynot represent the true duration of the event. In addition to theobservable times, there may be a latent waiting time 114 for theactivity and a latent service time 116 for the activity. Latent waitingtime 114 is defined as the time period from passing control to theactivity to the first event of the activity. Latent service time 116 isdefined as the time period from the last event of the activity topassing control to another activity. In addition to the observed waitingand service times, these times form the actual time consumed by anactivity. This allows the transition time to be split between the sourceand destination activities.

Referring now to FIG. 2, a second timeline 200 is shown. Rather thankeeping accurate timing information for the assignment, start, and stopof the events, in this second timeline 200 only the starting times 202are recorded. As a result, all that is observable is the transition time204 from one start time 202 to the next. The transition time is againdivided into a latent waiting time 206 and a latent service time 208.The present embodiments use a gamma distribution to estimate the latentwaiting time 206 and latent service time 208.

Referring now to FIG. 3, a method for finding the latent waiting andservice times is shown. Block 302 extracts transition times betweenactivities from an event log. As noted above, the event log isincomplete and shows only start-events or only end-events. Block 304then estimates the latent times from the transition times using, forexample, a gamma distribution. This information can then be used tooptimize the process, providing improvements for future activities toreduce latent waiting time and latent service time. For example, if thewaiting time of an activity is too long, then additional resources maybe provisioned to reduce the length of the queue. If the service time ofthe activity is too long, it may be possible to simplify the taskinvolved to reduce the amount of time needed to complete it.

Referring now to FIG. 4, a method for extracting the transition timesbetween activities is shown. Block 402 selects a new event from an eventlog. As noted above, the event log includes only one of start-events andend-events, signaling the beginning or end of an activity, respectively.Other events may be present in the event log, however, so the presentembodiments cycle through the events until the activity changes (i.e.,until a start-event or end-event for a new activity comes up). Thus, ifthe activity for the event selected by block 402 is the same as theactivity for the previous event at block 404, block 402 selects a newevent. If not, block 406 finds the transition time from the previousactivity change to the present event.

The present embodiments use a gamma distribution model to determinelatent waiting times and latent service times from the transition times.The gamma distribution is a sum of exponential distributions, which interm is used to model a time duration of a single activity and lifetime.An exponential distribution alone cannot fit actual duration data formost practical activities because the actual activities are composed ofseveral small activities. For example, underwriting in insuranceincludes reviewing an incoming application, measuring risk exposure, anddetermining the premium. By combining exponential distributions, thegamma distribution has enough flexibility to model the time duration ofsuch complex activities.

A gamma distribution is a two-parameter family of continuous probabilitydistributions. Its probability density function over a probabilisticvariable X>0 is defined by:

${p( {{X;l},a} )} = {\frac{X^{l - 1}}{{\Gamma (l)}\alpha^{l}}e^{- \frac{X}{\alpha}}}$

with a shape parameter l>0 and a scale parameter α>0, where Γ(x) is thegamma function Γ(x)=∫₀ ^(∞)s^(x-1) e^(−s) ds. X˜Gamma(l, α) if aprobabilistic variable X has this distribution.

Given an activity transition time log TL, as generated by FIG. 4 above,the random variable of latent service time of a source activity a isS_(a) and the random variable of latent waiting time of a destinationactivity b is W_(b). It is assumed that S_(a) follows a probabilisticdistribution Gamma(l_(a), α_(a)), where l_(a)>0, α_(a)>0, and that W_(b)follows a probabilistic distribution Gamma(m_(b), β_(b)), where m_(b)>0,β_(b)>0. This assumption is based on the Markov property, such that theprobability of time is independent from past events of the processinstance, whereas several process mining methods use past events. Sincelatent variable estimation imposes complex modeling and mathematics, theproblem definition is simplified herein.

The transition time from source activity a to destination activity b isT_(ab). Based on the above assumption, T_(ab)=S_(a)+W_(b). Theprobabilistic density function of T_(ab) can be obtained from aconvolutional integration of p(S_(a); l_(a), α_(a))*p(W; m_(b), β_(b))as:

$\begin{matrix}{{{P( {{T_{ab};l_{a}},\alpha_{a},m_{b},\beta_{b}} )} = {\int_{0}^{T_{ab}}{{p( {{S_{a};l_{a}},\alpha_{a}} )}{p( {{{T_{ab} - S_{a}};m_{b}},\beta_{b}} )}{dS}_{a}}}}\ } \\{= {\frac{e^{- \frac{(T_{ab})}{\beta_{b}}}}{{\Gamma ( l_{a} )}{\Gamma ( m_{b} )}\alpha_{a}^{l_{a}}\beta_{b}^{m_{b}}}{f( {T_{ab},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}\end{matrix}$ where${f( {t,l,\alpha,m,\beta} )} = {\int_{0}^{t}{{z^{l - 1}( {t - z} )}^{m - 1}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}z}\mspace{20mu} {dz}}}$

In one example, the latent time is not shared with other transitiontime. In that case, a single gamma distribution cannot be decomposedinto two gamma distributions because the distribution has thereproductive property. For example, if X˜Gamma(2, θ) and Y˜Gamma(3, θ),then X+Y follows Gamma(5, θ).

It can be assumed that the transition time from a source activity todifferent destination activities shares the same latent service time atthe transition source. In addition, transition times from differentsources to one destination share the same latent waiting time at thattransition destination. The latent services can be estimated if theinverse problem is solved.

Given an activity transition time log TL, the average latent waitingtime w _(a), the average latent service time s _(a), and the averageconsuming time c _(a), of the activity a, the first step is an inferenceof model parameters of p (T_(ab); l_(a), α_(a), m_(b), β_(b)). In theseembodiments, l_(a) is a shape parameter, α_(s) is a scale parameter of agamma distribution of latent service time of a, m_(a) is a shapeparameter, and β_(a) is a scale parameter of a gamma distribution oflatent waiting time for the activity a. Estimators {circumflex over(l)}_(a), {circumflex over (α)}_(a), {circumflex over (m)}_(b), and{circumflex over (β)}_(a) for the activity a are obtained and theestimator of s _(a) is then obtained as {circumflex over(l)}_(a){circumflex over (α)}_(a), the estimator of w _(a) can beobtained as {circumflex over (m)}_(b)β_(a), and the estimator of c _(a)can be obtained as the {circumflex over(s)}_(a)+duration_(a)+{circumflex over (w)}_(a). An average time isdefined as a triplet AT=(s, w, c), where s is a set of estimated averagelatent service times, w is a set of estimated average latent waitingtimes, and c is a set of estimated average consuming time.

Maximum likelihood estimation (MLE) is used to find estimators that makethe observed data most probable. Assuming the latent waiting time andthe latent service time follow a gamma distribution, the log likelihoodfunction becomes:

$\begin{matrix}{{\log \mspace{14mu} L} =} & {{\sum\limits_{{({a,b})} \in {Trans}}{\log \mspace{14mu} {p( {{t_{abk};l_{i}},\alpha_{i},m_{j},\beta_{j}} )}}}} \\{=} & {{\sum\limits_{{({a,b})} \in {Trans}}{\sum\limits_{k = 1}^{n_{ab}}\; {\log \frac{e^{- \frac{t_{abk}}{\beta_{b}}}}{{\Gamma ( l_{a} )}{\Gamma ( m_{b} )}\alpha_{a}^{l_{a}}\beta_{b}^{m_{b}}}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}}}\end{matrix}$

where t_(abk) is the k^(th) observed transition time from the sourceactivity a to the destination activity b and n_(ab) is the number ofobserved transition times from a to b. The parameters that give themaximization cannot be solved explicitly with equation transformationsbecause the latent probabilistic model makes the log likelihood functiontoo complex. In this formulation, “Trans” is the set of transitionsbetween activities.

Expectation maximization (EM) is used to find maximum likelihoodsolutions for probabilistic models having latent variables. In thiscase, the observed variables are T_(ab) and the latent variables areS_(a). The latent waiting time is removed from consideration by therelationship W_(b)=T_(ab)−S_(a). EM maximizes the likelihood iterativelywith the expectation and maximization steps.

In the expectation step a distribution q_(a)(S_(a)) is defined over thelatent variable S_(a). For any choice of q_(a)(S_(a)), the loglikelihood function is decomposed to:

log   p(t_(abk); l_(i), α_(i), m_(j), β_(j)) = ℒ(q_(a)(S_(a)), t_(abk)) + D_(KL)(q_(a)(S_(a))p(S_(a)t_(abk)))     where$\mspace{76mu} {{\mathcal{L}( {{q_{a}( S_{a} )},t_{abk}} )} = {\int_{0}^{t_{abk}}{{q_{a}( S_{a} )}\mspace{14mu} \log \frac{p( {t_{abk},{S_{a};l_{a}},\alpha_{a},m_{b},\beta_{b}} )}{q_{a}( S_{a} )}{dS}_{a}}}}$$D_{KL}( {{{q_{a}( S_{a} )}{}{p( {S_{a}t_{abk}} )}} = {\int_{0}^{t_{abk}}{{q_{a}( S_{a} )}\mspace{14mu} \log \frac{q_{a}( S_{a} )}{p( {t_{abk},{S_{a};l_{a}},\alpha_{a},m_{b},\beta_{b}} )}{dS}_{a}}}} $

Choosing p(t_(abk), S_(a); l_(a), α_(a), m_(b), β_(b))g as q_(a)(S_(a)),the Kullback-Leibler divergence goes to zero and

(q_(a)(S_(a)),t_(abk)) equals the log likelihood function.

In the maximization step, the parameters that maximize

(q_(a)(S_(a)),t_(abk)) are found, holding q_(a) (S_(a))=p(S_(a)|t_(abk);l_(a), α_(a), m_(b), β_(b)) are determined. The updated parameters arel′_(a), α′_(a), m′_(a), and β′_(a). By vanishing a constant part −∫₀^(t) ^(abk) q_(a)(S_(a))log q_(a) (S_(a))dS_(a) from

(q_(a)(S_(a)), t_(abk)), maximizing

(q_(a)(S_(a)), t_(abk)) is equivalent to maximizing:

$Q = {\sum\limits_{{({a,b})} \in {Trans}}{\sum\limits_{k = 1}^{n_{ab}}\; {\int_{0}^{t_{abk}}{{p( {{{S_{a}t_{abk}};l_{a}},\alpha_{a},m_{b},\beta_{b}} )}\mspace{14mu} \log \mspace{14mu} {p( {t_{abk},{S_{a};l_{a}^{\prime}},\alpha_{a}^{\prime},m_{b}^{\prime},\beta_{b}^{\prime}} )}{dS}_{a}}}}}$

By substituting the joint probability

${p( {T,{S;l},\alpha,m,\beta} )} = {{{p( {{S;l},\alpha} )}{p( {{{T - S};m},\beta} )}} = {\frac{{S^{l - 1}( {T - s} )}^{m - 1}}{{\Gamma (l)}{\Gamma (m)}\alpha^{l}\beta^{m}}e^{- \frac{T}{\beta}}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}S}}}$

and the posterior probability

${p( {{{ST};l},\alpha,m,\beta} )} = {\frac{p( {S,T} )}{p(T)} = {\frac{{S^{l - 1}( {T - s} )}^{m - 1}}{f( {T,l,\alpha,m,\beta} )}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}S}}}$

into Q produces:

$Q = {\sum\limits_{{({a,b})} \in {Trans}}{\sum\limits_{k = 1}^{n_{ab}}\{ {{- \frac{t_{abk}}{\beta_{b}^{\prime}}} - {\log \; {\Gamma ( l_{a}^{\prime} )}} - {\log \; {\Gamma ( m_{b}^{\prime} )}} - {l_{a}^{\prime}\mspace{14mu} \log \mspace{14mu} \alpha_{a}^{\prime}} - {m_{b}^{\prime}\mspace{14mu} \log \mspace{14mu} \beta_{b}^{\prime}} + {( {\frac{1}{\beta_{b}^{\prime}} - \frac{1}{\alpha_{a}^{\prime}}} )( \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l,\alpha_{a},m_{b},\beta_{b}} )} )} + {( {l_{a}^{\prime} - 1} )( \frac{g( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )} )} + {( {m_{b}^{\prime} - 1} )( \frac{h( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )} )}} \}}}$     where$\mspace{76mu} {{g( {t,l,\alpha,m,\beta} )} = {\int_{0}^{t}{{z^{l - 1}( {t - z} )}^{m - 1}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}z}\mspace{14mu} \log \mspace{14mu} z\mspace{14mu} {dz}}}}$$\mspace{76mu} {{h( {t,l,\alpha,m,\beta} )} = {\int_{0}^{t}{{z^{l - 1}( {t - z} )}^{m - 1}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}z}\mspace{14mu} {\log ( {t - z} )}\mspace{14mu} {dz}}}}$

The optimal point that gives the maximal value of Q satisfies:

$\mspace{76mu} {\frac{\partial Q}{\partial\alpha_{a}^{\prime}} = {{{{- \frac{l_{a}^{\prime}}{\alpha_{a}^{\prime}}}{\sum\limits_{b \in A_{d}}n_{ab}}} + {\frac{1}{\alpha_{a}^{\prime^{2}}}{\sum\limits_{b \in A_{d}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}}} = 0}}$$\frac{\partial Q}{\partial\beta_{a}^{\prime}} = {{{{- \frac{l_{a}^{\prime}}{\beta_{a}^{\prime}}}{\sum\limits_{a \in A_{s}}n_{ab}}} + {\frac{1}{\beta_{a}^{\prime^{2}}}{\sum\limits_{a \in A_{s}}{\sum\limits_{k = 1}^{n_{ab}}( {t_{abk} - \; \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}} )}}}} = 0}$$\frac{\partial Q}{\partial l_{a}^{\prime}} = {{{{- ( {{\psi ( l_{a}^{\prime} )} - {\log \mspace{14mu} \alpha_{a}^{\prime}}} )}{\sum\limits_{b \in A_{d}}n_{ab}}} + {\sum\limits_{b \in A_{d}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{g( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}} = 0}$$\frac{\partial Q}{\partial m_{a}^{\prime}} = {{{{- ( {{\psi ( m_{b}^{\prime} )} - {\log \mspace{14mu} \beta_{b}^{\prime}}} )}{\sum\limits_{a \in A_{s}}n_{ab}}} + {\sum\limits_{a \in A_{s}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{h( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}} = 0}$

where A_(s) is the set of source activities, A_(d) is the set ofdestination activities, and ψ(x) is the logarithmic derivative of thegamma function (known as the digamma function):

${\psi (x)} = {{\frac{d}{dx}\ln \mspace{14mu} {\Gamma (x)}} = {\frac{d\; {\Gamma (x)}}{dx}\text{/}{\Gamma (x)}}}$

By substituting

$\frac{\partial Q}{\partial\alpha_{a}^{\prime}}$

into the above equation for

${\frac{\partial Q}{\partial l_{a}^{\prime}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial Q}{\partial\beta_{a}^{\prime}}\mspace{14mu} {into}\mspace{14mu} \frac{\partial Q}{\partial m_{a}^{\prime}}},$

the following nonlinear equations are produced:

${{\psi ( l_{a}^{\prime} )} - {\log ( l_{a}^{\prime} )} - {\frac{1}{\Sigma_{b \in A_{d}}\mspace{14mu} n_{ab}}{\sum\limits_{b \in A_{d}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{g( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}} + {\log ( {\frac{1}{\Sigma_{b \in A_{d}}\mspace{14mu} n_{ab}}{\sum\limits_{b \in A_{d}}{\sum\limits_{k = 1}^{n_{ab}}\frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}} )}} = 0$${{\psi ( m_{b}^{\prime} )} - {\log ( m_{b}^{\prime} )} - {\frac{1}{\Sigma_{a \in A_{s}}\mspace{14mu} n_{ab}}{\sum\limits_{a \in A_{s}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{h( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}} + {\log ( {\frac{1}{\Sigma_{a \in A_{s}}\mspace{14mu} n_{ab}}{\sum\limits_{a \in A_{s}}{\sum\limits_{k = 1}^{n_{ab}}( {t_{abk} - \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}} )}}} )}} = 0$

The optimal parameters, l′_(a) and m′_(b), are given by solving thesenonlinear, one-variable equations. By substituting the parameters backinto

${\frac{\partial Q}{\partial\alpha_{a}^{\prime}}\mspace{14mu} {and}\mspace{14mu} \frac{\partial Q}{\partial\beta_{a}^{\prime}}},$

the optimal parameters α′_(a) and β′_(b) are produced:

$\alpha_{a}^{\prime} = {\frac{1}{l_{a}^{\prime}\mspace{14mu} \Sigma_{b \in A_{d}}\mspace{14mu} n_{ab}}{\sum\limits_{b \in A_{d}}{\sum\limits_{k = 1}^{n_{ab}}\; \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}}$$\beta_{b}^{\prime} = {\frac{1}{m_{b}^{\prime}\mspace{14mu} \Sigma_{a \in A_{s}}\mspace{14mu} n_{ab}}{\sum\limits_{a \in s}{\sum\limits_{k = 1}^{n_{ab}}( {t_{abk} - \; \frac{f( {t_{abk},{l_{a} + 1},\alpha_{a},m_{b},\beta_{b}} )}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}} )}}}$

The latent service time is then calculated as s _(a)=l_(a)α_(a), thelatent waiting time is calculated as w _(a)=m_(a)β_(a), and theconsuming time is calculated as c _(a)=lw_(a)+duration_(a)+ls_(a).

Referring now to FIG. 5, a method for extracting the latent times fromthe transition times is shown. The present embodiment iterativelyconsiders transition times from source activities to destinationactivities until a log likelihood function converges. Block 501initializes the parameters, with the likelihood L being set to −∞, andeach of the parameters l_(a), α_(a), m_(b), and β_(b) each being set to(Σ_((a,b)∈Trans)Σ_(k=0) ^(n) ^(ab) t_(abk))/(Σ_((a,b)∈Trans)n_(ab)).

Block 502 determines l_(a) for each source activity, block 504determines α_(a) for each source activity, block 506 determines m_(b)for each destination activity, and block 508 determines β_(b) for eachdestination activity. Using these parameters, a likelihood L iscalculated as described above in block 510. Block 512 determines whetherthe likelihood has converged. If not, processing returns to block 502.If so, block 514 determines the latent service time and latent waitingtime using the determined parameters. As the method iterates, theparameters from each iteration are used in the next iteration to formthe basis for the next likelihood calculation. Eventually the likelihoodwill stabilize, as determined by a difference between two consecutivelikelihoods being within a threshold value of one another.

The present invention may be a system, a method, and/or a computerprogram product. The computer program product may include a computerreadable storage medium (or media) having computer readable programinstructions thereon for causing a processor to carry out aspects of thepresent invention.

The computer readable storage medium can be a tangible device that canretain and store instructions for use by an instruction executiondevice. The computer readable storage medium may be, for example, but isnot limited to, an electronic storage device, a magnetic storage device,an optical storage device, an electromagnetic storage device, asemiconductor storage device, or any suitable combination of theforegoing. A non-exhaustive list of more specific examples of thecomputer readable storage medium includes the following: a portablecomputer diskette, a hard disk, a random access memory (RAM), aread-only memory (ROM), an erasable programmable read-only memory (EPROMor Flash memory), a static random access memory (SRAM), a portablecompact disc read-only memory (CD-ROM), a digital versatile disk (DVD),a memory stick, a floppy disk, a mechanically encoded device such aspunch-cards or raised structures in a groove having instructionsrecorded thereon, and any suitable combination of the foregoing. Acomputer readable storage medium, as used herein, is not to be construedas being transitory signals per se, such as radio waves or other freelypropagating electromagnetic waves, electromagnetic waves propagatingthrough a waveguide or other transmission media (e.g., light pulsespassing through a fiber-optic cable), or electrical signals transmittedthrough a wire.

Computer readable program instructions described herein can bedownloaded to respective computing/processing devices from a computerreadable storage medium or to an external computer or external storagedevice via a network, for example, the Internet, a local area network, awide area network and/or a wireless network. The network may comprisecopper transmission cables, optical transmission fibers, wirelesstransmission, routers, firewalls, switches, gateway computers and/oredge servers. A network adapter card or network interface in eachcomputing/processing device receives computer readable programinstructions from the network and forwards the computer readable programinstructions for storage in a computer readable storage medium withinthe respective computing/processing device.

Computer readable program instructions for carrying out operations ofthe present invention may be assembler instructions,instruction-set-architecture (ISA) instructions, machine instructions,machine dependent instructions, microcode, firmware instructions,state-setting data, or either source code or object code written in anycombination of one or more programming languages, including an objectoriented programming language such as Smalltalk, C++ or the like, andconventional procedural programming languages, such as the “C”programming language or similar programming languages. The computerreadable program instructions may execute entirely on the user'scomputer, partly on the user's computer, as a stand-alone softwarepackage, partly on the user's computer and partly on a remote computeror entirely on the remote computer or server. In the latter scenario,the remote computer may be connected to the user's computer through anytype of network, including a local area network (LAN) or a wide areanetwork (WAN), or the connection may be made to an external computer(for example, through the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example,programmable logic circuitry, field-programmable gate arrays (FPGA), orprogrammable logic arrays (PLA) may execute the computer readableprogram instructions by utilizing state information of the computerreadable program instructions to personalize the electronic circuitry,in order to perform aspects of the present invention.

Aspects of the present invention are described herein with reference toflowchart illustrations and/or block diagrams of methods, apparatus(systems), and computer program products according to embodiments of theinvention. It will be understood that each block of the flowchartillustrations and/or block diagrams, and combinations of blocks in theflowchart illustrations and/or block diagrams, can be implemented bycomputer readable program instructions.

These computer readable program instructions may be provided to aprocessor of a general purpose computer, special purpose computer, orother programmable data processing apparatus to produce a machine, suchthat the instructions, which execute via the processor of the computeror other programmable data processing apparatus, create means forimplementing the functions/acts specified in the flowchart and/or blockdiagram block or blocks. These computer readable program instructionsmay also be stored in a computer readable storage medium that can directa computer, a programmable data processing apparatus, and/or otherdevices to function in a particular manner, such that the computerreadable storage medium having instructions stored therein comprises anarticle of manufacture including instructions which implement aspects ofthe function/act specified in the flowchart and/or block diagram blockor blocks.

The computer readable program instructions may also be loaded onto acomputer, other programmable data processing apparatus, or other deviceto cause a series of operational steps to be performed on the computer,other programmable apparatus or other device to produce a computerimplemented process, such that the instructions which execute on thecomputer, other programmable apparatus, or other device implement thefunctions/acts specified in the flowchart and/or block diagram block orblocks.

The flowchart and block diagrams in the Figures illustrate thearchitecture, functionality, and operation of possible implementationsof systems, methods, and computer program products according to variousembodiments of the present invention. In this regard, each block in theflowchart or block diagrams may represent a module, segment, or portionof instructions, which comprises one or more executable instructions forimplementing the specified logical function(s). In some alternativeimplementations, the functions noted in the block may occur out of theorder noted in the figures. For example, two blocks shown in successionmay, in fact, be executed substantially concurrently, or the blocks maysometimes be executed in the reverse order, depending upon thefunctionality involved. It will also be noted that each block of theblock diagrams and/or flowchart illustration, and combinations of blocksin the block diagrams and/or flowchart illustration, can be implementedby special purpose hardware-based systems that perform the specifiedfunctions or acts or carry out combinations of special purpose hardwareand computer instructions.

Reference in the specification to “one embodiment” or “an embodiment” ofthe present principles, as well as other variations thereof, means thata particular feature, structure, characteristic, and so forth describedin connection with the embodiment is included in at least one embodimentof the present principles. Thus, the appearances of the phrase “in oneembodiment” or “in an embodiment”, as well any other variations,appearing in various places throughout the specification are notnecessarily all referring to the same embodiment.

It is to be appreciated that the use of any of the following “/”,“and/or”, and “at least one of”, for example, in the cases of “A/B”, “Aand/or B” and “at least one of A and B”, is intended to encompass theselection of the first listed option (A) only, or the selection of thesecond listed option (B) only, or the selection of both options (A andB). As a further example, in the cases of “A, B, and/or C” and “at leastone of A, B, and C”, such phrasing is intended to encompass theselection of the first listed option (A) only, or the selection of thesecond listed option (B) only, or the selection of the third listedoption (C) only, or the selection of the first and the second listedoptions (A and B) only, or the selection of the first and third listedoptions (A and C) only, or the selection of the second and third listedoptions (B and C) only, or the selection of all three options (A and Band C). This may be extended, as readily apparent by one of ordinaryskill in this and related arts, for as many items listed.

Referring now to FIG. 6, a system 600 for estimating latent service andwaiting times is shown. The system 600 includes a hardware processor 602and memory 604. The system 600 also includes a set of functionalmodules. These modules may be implemented as software that is stored inmemory 604 and executed on the processor 602 or may, alternatively, beimplemented as one or more discrete hardware components in the form of,e.g., application specific integrated chips or field programmable gatearrays.

The memory stores an event log 606 that records partial eventinformation in the memory 604 of an executed activity. For example, theevent log 606 may record only start-event information or end-eventinformation. Based on the event log, a transition time module 608determines a transition time between different activities. This createsa transition time log. A parameter module 610 determines a set of latenttime estimation parameters based on the transition time log and latenttime module 612 produces an estimate of the latent waiting time andlatent service time for each activity.

It should be noted in particular that the present embodiments may beemployed to improve the functioning of a computer system. In particular,computing system logs often track only the start or stop times of acomputer process, providing little insight as to how the interveningtime is used. By determining latent service and wait times, optimizationof processes can be performed and the performance of the computingsystem can therefore be substantially improved. The present embodimentstherefore represent a significant improvement to the computer systemitself, improving the speed of its processes.

Having described preferred embodiments of estimation of latent waitingand service times from incomplete event logs (which are intended to beillustrative and not limiting), it is noted that modifications andvariations can be made by persons skilled in the art in light of theabove teachings. It is therefore to be understood that changes may bemade in the particular embodiments disclosed which are within the scopeof the invention as outlined by the appended claims. Having thusdescribed aspects of the invention, with the details and particularityrequired by the patent laws, what is claimed and desired protected byLetters Patent is set forth in the appended claims.

1. A method for estimating latent service and latent wait times,comprising: estimating parameters that characterize latent waiting timeand latent service time for each of a plurality of activities based ontransition times between the plurality of activities; estimating alatent waiting time and latent service time for each activity based onthe estimated parameters using a processor; and provisioning computingresources to reduce the latent waiting time.
 2. The method of claim 1,further comprising determining a likelihood that the estimatedparameters would reproduce the partial event log.
 3. The method of claim2, further comprising iterating the estimation of parameters and thedetermination of the likelihood until the likelihood converges.
 4. Themethod of claim 3, wherein the iteration ends when a likelihood from aprevious iteration is within a threshold value of a likelihood from acurrent iteration.
 5. The method of claim 2, wherein the likelihood isdetermined by:$\sum\limits_{{({a,b})} \in {Trans}}{\sum\limits_{k = 1}^{n_{ab}}{\log \frac{e^{- \frac{t_{abk}}{\beta_{b}}}}{{\Gamma ( l_{a} )}{\Gamma ( m_{b} )}\alpha_{a}^{l_{a}}\beta_{b}^{m_{b}}}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}$wherein a is a source activity, b is a destination activity, n_(ab) is anumber of observed transition times from a to b, t_(abk) is the k^(th)transition time from a to b, l_(a), α_(a), m_(b), and β_(b) areparameters of the gamma function Γ(x), and${f( {t,l,\alpha,m,\beta} )} = {\int_{0}^{t}{{z^{l - 1}( {t - z} )}^{m - 1}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}z}\mspace{14mu} {{dz}.}}}$6. The method of claim 1, wherein the partial event log comprises onlyone of start-event information and end-event information for eachactivity.
 7. The method of claim 1, wherein extracting transition timesbetween activities comprises: determining a first event at which asource activity is first recorded in the partial event log; anddetermining a second event at which a destination activity is firstrecorded in the partial event log, wherein the transition time betweenthe source activity and the destination activity is a difference in timebetween the first event and the second event.
 8. The method of claim 1,wherein the latent service time is estimated as:s _(a)=l_(a)α_(a) wherein a is an activity, l_(a) is a shape parameterof a gamma distribution for the activity, and α_(a) is a scale parameterof the gamma distribution for the activity.
 9. The method of claim 1,wherein the latent waiting time is estimated as:w _(a)=m_(a)β_(a) wherein a is an activity, m_(a) is a shape parameterof a gamma distribution for the activity, and β_(a) is a scale parameterof the gamma distribution for the activity.
 10. The method of claim 1,wherein latent waiting time represents a time period from passingcontrol to an activity to the first event of the activity, and whereinlatent service time represents a time period from a last event of anactivity to passing control to another activity.
 11. A computer readablestorage medium comprising a computer readable program for estimatinglatent service and latent wait times, wherein the computer readableprogram when executed on a computer causes the computer to perform thesteps of: estimating parameters that characterize latent waiting timeand latent service time for each of a plurality of activities based ontransition times between the plurality of activities; estimating alatent waiting time and latent service time for each activity based onthe estimated parameters using a processor; and provisioning computingresources to reduce the latent waiting time.
 12. A system for estimatinglatent service and latent wait times, comprising: a parameter moduleconfigured to estimate parameters that characterize latent waiting timeand latent service time for each of a plurality of activities based ontransition times between the plurality of activities; and a latent timemodule comprising a processor configured to estimate a latent waitingtime and latent service time for each activity based on the estimatedparameters, and to provision computing resources to reduce the latentwaiting time.
 13. The system of claim 12, wherein the parameter moduleis further configured to determine a likelihood that the estimatedparameters would reproduce the partial event log.
 14. The system ofclaim 13, wherein the parameter module is further configured to iteratethe estimation of parameters and the determination of the likelihooduntil the likelihood converges.
 15. The system of claim 14, wherein theparameter module is further configured to halt iteration when alikelihood from a previous iteration is within a threshold value of alikelihood from a current iteration.
 16. The system of claim 13, whereinthe likelihood is determined by:$\sum\limits_{{({a,b})} \in {Trans}}{\sum\limits_{k = 1}^{n_{ab}}{\log \frac{e^{- \frac{t_{abk}}{\beta_{b}}}}{{\Gamma ( l_{a} )}{\Gamma ( m_{b} )}\alpha_{a}^{l_{a}}\beta_{b}^{m_{b}}}{f( {t_{abk},l_{a},\alpha_{a},m_{b},\beta_{b}} )}}}$wherein a is a source activity, b is a destination activity, n_(ab) is anumber of observed transition times from a to b, t_(abk) is the k^(th)transition time from a to b, l_(a), α_(a), m_(b), and β_(b) areparameters of the gamma function Γ(x), and${f( {t,l,\alpha,m,\beta} )} = {\int_{0}^{t}{{z^{l - 1}( {t - z} )}^{m - 1}e^{{({\frac{1}{\beta} - \frac{1}{\alpha}})}z}\mspace{14mu} {{dz}.}}}$17. The system of claim 12, wherein the partial event log comprises onlyone of start-event information and end-event information for eachactivity.
 18. The system of claim 12, wherein the transition time moduleis further configured to determine a first event at which a sourceactivity is first recorded in the partial event log and to determine asecond event at which a destination activity is first recorded in thepartial event log, wherein the transition time between the sourceactivity and the destination activity is a difference in time betweenthe first event and the second event.
 19. The system of claim 12,wherein the latent service time is estimated as:s _(a)=l_(a)α_(a) wherein a is an activity, l_(a) is a shape parameterof a gamma distribution for the activity, and α_(a) is a scale parameterof the gamma distribution for the activity.
 20. The system of claim 12,wherein the latent waiting time is estimated as:w _(a)=m_(a)β_(a) wherein a is an activity, m_(a) is a shape parameterof a gamma distribution for the activity, and β_(a) is a scale parameterof the gamma distribution for the activity.